diff --git a/README.html b/README.html index f419568..4c55265 100644 --- a/README.html +++ b/README.html @@ -157,7 +157,7 @@ GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see http://www.gnu.org/licenses/.

-

Access a web-copy of the notes for printing

+

Access a PDF render of the notes (It is recommended to refer to this instead of the GitHub rendered page!)

I accept pull requests or suggestions but the content must not be copyrighted under a non-GPL compatible license.

Fourier transform identities

@@ -170,18 +170,18 @@ along with this program. If not, see http - + - + - + @@ -189,8 +189,8 @@ along with this program. If not, see http - - + + @@ -214,7 +214,7 @@ along with this program. If not, see http - + @@ -250,7 +250,7 @@ along with this program. If not, see http - + @@ -262,36 +262,14 @@ along with this program. If not, see http
rect(tT)Π(tT)\text{rect}\left(\frac{t}{T}\right)\quad\Pi\left(\frac{t}{T}\right)T sinc(fT)T \, \text{sinc}(fT)Tsinc(fT)T \text{sinc}(fT)
sinc(2Wt)\text{sinc}(2Wt) 12Wrect(f2W)12WΠ(f2W)\frac{1}{2W}\text{rect}\left(\frac{f}{2W}\right)\quad\frac{1}{2W}\Pi\left(\frac{f}{2W}\right)
exp⁡(−at)u(t), a>0\exp(-at)u(t), \, a>0exp⁡(−at)u(t),a>0\exp(-at)u(t),\quad a>0 1a+j2πf\frac{1}{a + j2\pi f}
exp⁡(−a∣t∣), a>0\exp(-a\lvert t \rvert), \, a>0exp⁡(−a∣t∣),a>0\exp(-a\lvert t \rvert),\quad a>0 2aa2+(2πf)2\frac{2a}{a^2 + (2\pi f)^2}
exp⁡(−πf2)\exp(-\pi f^2)
{1−∣t∣T,∣t∣<T0,∣t∣≥T\begin{cases} 1 - \frac{\lvert t \rvert}{T}, & \lvert t \rvert < T \\ 0, & \lvert t \rvert \geq T \end{cases}T sinc2(fT)T \, \text{sinc}^2(fT)1−∣t∣T,∣t∣<T1 - \frac{\lvert t \rvert}{T},\quad\lvert t \rvert < TTsinc2(fT)T \text{sinc}^2(fT)
δ(t)\delta(t)
g(bt−a)g(bt-a)1∣b∣exp⁡(−j2πa(f/b))⋅G(f/b)shift & scale\frac{1}{|b|}\exp(-j2\pi a(f/b))\cdot G(f/b)\quad\text{shift \& scale}1∣b∣exp⁡(−j2πa(f/b))⋅G(f/b)shift and scale\frac{1}{|b|}\exp(-j2\pi a(f/b))\cdot G(f/b)\quad\text{shift and scale}
ddtg(t)\frac{d}{dt}g(t)
1πt\frac{1}{\pi t}−j sgn(f)-j \, \text{sgn}(f)−jsgn(f)-j \text{sgn}(f)
u(t)u(t)
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Function NameFormula
Unit Step Functionu(t)={1,t>012,t=00,t<0u(t) = \begin{cases} 1, & t > 0 \\ \frac{1}{2}, & t = 0 \\ 0, & t < 0 \end{cases}
Signum Functionsgn(t)={+1,t>00,t=0−1,t<0\text{sgn}(t) = \begin{cases} +1, & t > 0 \\ 0, & t = 0 \\ -1, & t < 0 \end{cases}
sinc Functionsinc(2Wt)=sin⁡(2πWt)2πWt\text{sinc}(2Wt) = \frac{\sin(2\pi W t)}{2\pi W t}
Rectangular Functionrect(t)=Π(t)={1,−0.5<t<0.50,∣t∣>0.5\text{rect}(t) = \Pi(t) = \begin{cases} 1, & -0.5 < t < 0.5 \\ 0, & \lvert t \rvert > 0.5 \end{cases}
Convolutiong(t)∗h(t)=(g∗h)(t)=∫∞∞g(τ)h(t−τ)dτg(t)*h(t)=(g*h)(t)=\int_\infty^\infty g(\tau)h(t-\tau)d\tau
+

u(t)={1,t>012,t=00,t<0Unit Step Functionsgn(t)={+1,t>00,t=0−1,t<0Signum Functionsinc(2Wt)=sin⁡(2πWt)2πWtsinc Functionrect(t)=Π(t)={1,−0.5<t<0.50,∣t∣>0.5Rectangular/Gate Functiong(t)∗h(t)=(g∗h)(t)=∫∞∞g(τ)h(t−τ)dτConvolution\begin{align*} + u(t) &= \begin{cases} 1, & t > 0 \\ \frac{1}{2}, & t = 0 \\ 0, & t < 0 \end{cases}&\text{Unit Step Function}\\ + \text{sgn}(t) &= \begin{cases} +1, & t > 0 \\ 0, & t = 0 \\ -1, & t < 0 \end{cases}&\text{Signum Function}\\ + \text{sinc}(2Wt) &= \frac{\sin(2\pi W t)}{2\pi W t}&\text{sinc Function}\\ + \text{rect}(t) = \Pi(t) &= \begin{cases} 1, & -0.5 < t < 0.5 \\ 0, & \lvert t \rvert > 0.5 \end{cases}&\text{Rectangular/Gate Function}\\ + g(t)*h(t)=(g*h)(t)&=\int_\infty^\infty g(\tau)h(t-\tau)d\tau&\text{Convolution}\\ +\end{align*} +

Fourier transform of continuous time periodic signal

Required for some questions on sampling:

@@ -310,7 +288,7 @@ along with this program. If not, see http

rect

Bessel function

∑n∈ZJn2(β)=1Jn(β)=(−1)nJ−n(β)\begin{align*} - \sum_{n\in\Z}{J_n}^2(\beta)&=1\\ + \sum_{n\in\mathbb{Z}}{J_n}^2(\beta)&=1\\ J_n(\beta)&=(-1)^nJ_{-n}(\beta) \end{align*}

@@ -381,7 +359,7 @@ G_y(f)&=G(f)G_w(f)\\ \sin(A-\pi/2)&=-\cos(A)\\ \cos(A-\pi/2)&=\sin(A)\\ \cos(A+\pi/2)&=-\sin(A)\\ - \int_{x\in\R}\text{sinc}(A x) &= \frac{1}{|A|}\\ + \int_{x\in\mathbb{R}}\text{sinc}(A x) &= \frac{1}{|A|}\\ \end{align*}

cos⁡(A+B)=cos⁡(A)cos⁡(B)−sin⁡(A)sin⁡(B)sin⁡(A+B)=sin⁡(A)cos⁡(B)+cos⁡(A)sin⁡(B)cos⁡(A)cos⁡(B)=12(cos⁡(A−B)+cos⁡(A+B))cos⁡(A)sin⁡(B)=12(sin⁡(A+B)−sin⁡(A−B))sin⁡(A)sin⁡(B)=12(cos⁡(A−B)−cos⁡(A+B))\begin{align*} @@ -468,9 +446,6 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\ s(t) &= A_c\cos\left[2\pi f_c t + \beta \sin(2\pi f_m t)\right] \Leftrightarrow s(t)= A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[2\pi(f_c+nf_m)t] \end{align*}

-

FM signal power

Pav=Ac22Pband_index=Ac2Jband_index2(β)2band_index=0  ⟹  fc+0fmband_index=1  ⟹  fc+1fm,…\begin{align*} P_\text{av}&=\frac{{A_c}^2}{2}\\ @@ -1236,7 +1211,7 @@ h400000v40h-400000z"/>​ 0.000.00 -$0.5 +0.50.5 2.302.30 0.0107240.010724 4.554.55 @@ -1708,11 +1683,11 @@ h400000v40h-400000z"/>​

+``` -->

ISI, channel model

Nyquist criterion for zero ISI

TODO:

@@ -1734,12 +1709,6 @@ $$ -->
  • Use the formula for DD below
  • Consult the BER table below to get the BER which relates the noise of the channel N0N_0 to EbE_b and to RbR_b.
    • - @@ -1771,7 +1740,7 @@ $$ --> \end{align*}

    Nyquist stuff

    -

    Condition for 0 ISI

    +

    Condition for 0 ISI TODO:

    Pr(kT)={1k=00k≠0P_r(kT)=\begin{cases} 1 & k=0\\ 0 & k\neq0 @@ -2039,8 +2008,7 @@ Horizontal, ​p11​p21​⋮pM1​​p12​p22​⋮pM2​​……⋱…​p1N​p2N​⋮pMN​​​

    P(yj∣xi)y1y2…yNx1p11p12…p1Nx2p21p22…p2N⋮⋮⋮⋱⋮xMpM1pM2…pMN\begin{array}{c|cccc} - P(y_j|x_i)& y_1 & y_2 & \dots & y_N \\ - \hline + P(y_j|x_i)& y_1 & y_2 & \dots & y_N \\\hline x_1 & p_{11} & p_{12} & \dots & p_{1N} \\ x_2 & p_{21} & p_{22} & \dots & p_{2N} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ @@ -2048,7 +2016,7 @@ M347 1759 V0 H263 V1759 v1800 v1759 h84z"/>

    Input has probability distribution pX(ai)=P(X=ai)p_X(a_i)=P(X=a_i)

    -

    Channel maps alphabet {a1,…,aM}→{b1,…,bN}\{a_1,\dots,a_M\} \to \{b_1,\dots,b_N\}

    +

    Channel maps alphabet ‘{a1,…,aM}→{b1,…,bN}‘`\{a_1,\dots,a_M\} \to \{b_1,\dots,b_N\}`

    Output has probabiltiy distribution pY(bj)=P(y=bj)p_Y(b_j)=P(y=b_j)

    pY(bj)=∑i=1MP[x=ai,y=bj]1≤j≤N=∑i=1MP[X=ai]P[Y=bj∣X=ai][pY(b0)pY(b1)…pY(bj)]=[pX(a0)pX(a1)…pX(ai)]×P\begin{align*} p_Y(b_j) &= \sum_{i=1}^{M}P[x=a_i,y=b_j]\quad 1\leq j\leq N \\ @@ -2131,7 +2099,7 @@ M347 1759 V0 H263 V1759 v1800 v1759 h84z"/>

    - +
    Minimum distance dmind_\text{min}IMPORTANT: x≠0x\neq\bold{0}, excludes weight of all-zero codeword. For a linear block code, dmin=wmind_\text{min}=w_\text{min}IMPORTANT: x≠0x\neq\textbf{0}, excludes weight of all-zero codeword. For a linear block code, dmin=wmind_\text{min}=w_\text{min}
    @@ -2245,8 +2213,7 @@ M347 1759 V0 H263 V1759 v1800 v1759 h84z"/>

    Example:

    x1x2x3x4x510110011110000011001\begin{array}{cccc} - x_1 & x_2 & x_3 & x_4 & x_5 \\ - \hline + x_1 & x_2 & x_3 & x_4 & x_5 \\\hline \color{magenta}1&\color{magenta}0&1&1&0\\ \color{magenta}0&\color{magenta}1&1&1&1\\ \color{magenta}0&\color{magenta}0&0&0&0\\ @@ -2254,16 +2221,15 @@ M347 1759 V0 H263 V1759 v1800 v1759 h84z"/>

    Set x1,x2x_1,x_2 as information bits. Express x3,x4,x5x_3,x_4,x_5 in terms of x1,x2x_1,x_2.

    -

    x3=x1⊕x2x4=x1⊕x2x5=x2  ⟹  P=x1x2x311x411x501H=[111101100010001]\begin{align*} -\begin{align*} +

    x3=x1⊕x2x4=x1⊕x2x5=x2  ⟹  P=x1x2x311x411x501H=[111101100010001]\begin{align*} +\begin{aligned} x_3 &= x_1\oplus x_2\\ x_4 &= x_1\oplus x_2\\ x_5 &= x_2\\ -\end{align*} +\end{aligned} \implies\textbf{P}&= -\begin{array}{c|ccc} - & x_1 & x_2 \\ - \hline +\begin{array}{c|cc} + & x_1 & x_2 \\\hline x_3&1&1&\\ x_4&1&1&\\ x_5&0&1&\\ @@ -2290,8 +2256,9 @@ M347 1759 V0 H263 V1759 v0 v1759 h84z"/>uu errors: dmin≥2u+1d_\text{min}\geq 2u+1

    CHECKLIST

    diff --git a/README.md b/README.md index 7e77b17..dadcc4d 100644 --- a/README.md +++ b/README.md @@ -25,45 +25,47 @@ along with this program. If not, see . -[Access a web-copy of the notes for printing](/README.html) +[Access a PDF render of the notes (**It is recommended to refer to this instead of the GitHub rendered page!**)](/README.pdf) I accept pull requests or suggestions but the content must not be copyrighted under a non-GPL compatible license. ## Fourier transform identities -| **Time Function** | **Fourier Transform** | -| --------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------- | -| $\text{rect}\left(\frac{t}{T}\right)\quad\Pi\left(\frac{t}{T}\right)$ | $T \, \text{sinc}(fT)$ | -| $\text{sinc}(2Wt)$ | $\frac{1}{2W}\text{rect}\left(\frac{f}{2W}\right)\quad\frac{1}{2W}\Pi\left(\frac{f}{2W}\right)$ | -| $\exp(-at)u(t), \, a>0$ | $\frac{1}{a + j2\pi f}$ | -| $\exp(-a\lvert t \rvert), \, a>0$ | $\frac{2a}{a^2 + (2\pi f)^2}$ | -| $\exp(-\pi t^2)$ | $\exp(-\pi f^2)$ | -| $\begin{cases} 1 - \frac{\lvert t \rvert}{T}, & \lvert t \rvert < T \\ 0, & \lvert t \rvert \geq T \end{cases}$ | $T \, \text{sinc}^2(fT)$ | -| $\delta(t)$ | $1$ | -| $1$ | $\delta(f)$ | -| $\delta(t - t_0)$ | $\exp(-j2\pi f t_0)$ | -| $g(t-a)$ | $\exp(-j2\pi fa)G(f)\quad\text{shift property}$ | -| $g(bt)$ | $\frac{G(f/b)}{\|b\|}\quad\text{scaling property}$ | -| $g(bt-a)$ | $\frac{1}{\|b\|}\exp(-j2\pi a(f/b))\cdot G(f/b)\quad\text{shift \& scale}$ | -| $\frac{d}{dt}g(t)$ | $j2\pi fG(f)\quad\text{differentiation property}$ | -| $G(t)$ | $g(-f)\quad\text{duality property}$ | -| $g(t)h(t)$ | $G(f)*H(f)$ | -| $g(t)*h(t)$ | $G(f)H(f)$ | -| $\exp(j2\pi f_c t)$ | $\delta(f - f_c)$ | -| $\cos(2\pi f_c t)$ | $\frac{1}{2}[\delta(f - f_c) + \delta(f + f_c)]$ | -| $\sin(2\pi f_c t)$ | $\frac{1}{2j} [\delta(f - f_c) - \delta(f + f_c)]$ | -| $\text{sgn}(t)$ | $\frac{1}{j\pi f}$ | -| $\frac{1}{\pi t}$ | $-j \, \text{sgn}(f)$ | -| $u(t)$ | $\frac{1}{2} \delta(f) + \frac{1}{j2\pi f}$ | -| $\sum_{n=-\infty}^{\infty} \delta(t - nT_0)$ | $\frac{1}{T_0} \sum_{n=-\infty}^{\infty} \delta\left(f - \frac{n}{T_0}\right)=f_0 \sum_{n=-\infty}^{\infty} \delta\left(f - n f_0\right)$ | +| **Time Function** | **Fourier Transform** | +| --------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------- | +| $\text{rect}\left(\frac{t}{T}\right)\quad\Pi\left(\frac{t}{T}\right)$ | $T \text{sinc}(fT)$ | +| $\text{sinc}(2Wt)$ | $\frac{1}{2W}\text{rect}\left(\frac{f}{2W}\right)\quad\frac{1}{2W}\Pi\left(\frac{f}{2W}\right)$ | +| $\exp(-at)u(t),\quad a>0$ | $\frac{1}{a + j2\pi f}$ | +| $\exp(-a\lvert t \rvert),\quad a>0$ | $\frac{2a}{a^2 + (2\pi f)^2}$ | +| $\exp(-\pi t^2)$ | $\exp(-\pi f^2)$ | +| $1 - \frac{\lvert t \rvert}{T},\quad\lvert t \rvert < T$ | $T \text{sinc}^2(fT)$ | +| $\delta(t)$ | $1$ | +| $1$ | $\delta(f)$ | +| $\delta(t - t_0)$ | $\exp(-j2\pi f t_0)$ | +| $g(t-a)$ | $\exp(-j2\pi fa)G(f)\quad\text{shift property}$ | +| $g(bt)$ | $\frac{G(f/b)}{\|b\|}\quad\text{scaling property}$ | +| $g(bt-a)$ | $\frac{1}{\|b\|}\exp(-j2\pi a(f/b))\cdot G(f/b)\quad\text{shift and scale}$ | +| $\frac{d}{dt}g(t)$ | $j2\pi fG(f)\quad\text{differentiation property}$ | +| $G(t)$ | $g(-f)\quad\text{duality property}$ | +| $g(t)h(t)$ | $G(f)*H(f)$ | +| $g(t)*h(t)$ | $G(f)H(f)$ | +| $\exp(j2\pi f_c t)$ | $\delta(f - f_c)$ | +| $\cos(2\pi f_c t)$ | $\frac{1}{2}[\delta(f - f_c) + \delta(f + f_c)]$ | +| $\sin(2\pi f_c t)$ | $\frac{1}{2j} [\delta(f - f_c) - \delta(f + f_c)]$ | +| $\text{sgn}(t)$ | $\frac{1}{j\pi f}$ | +| $\frac{1}{\pi t}$ | $-j \text{sgn}(f)$ | +| $u(t)$ | $\frac{1}{2} \delta(f) + \frac{1}{j2\pi f}$ | +| $\sum_{n=-\infty}^{\infty} \delta(t - nT_0)$ | $\frac{1}{T_0} \sum_{n=-\infty}^{\infty} \delta\left(f - \frac{n}{T_0}\right)=f_0 \sum_{n=-\infty}^{\infty} \delta\left(f - n f_0\right)$ | -| **Function Name** | **Formula** | -| -------------------- | ------------------------------------------------------------------------------------------------------- | -| Unit Step Function | $u(t) = \begin{cases} 1, & t > 0 \\ \frac{1}{2}, & t = 0 \\ 0, & t < 0 \end{cases}$ | -| Signum Function | $\text{sgn}(t) = \begin{cases} +1, & t > 0 \\ 0, & t = 0 \\ -1, & t < 0 \end{cases}$ | -| sinc Function | $\text{sinc}(2Wt) = \frac{\sin(2\pi W t)}{2\pi W t}$ | -| Rectangular Function | $\text{rect}(t) = \Pi(t) = \begin{cases} 1, & -0.5 < t < 0.5 \\ 0, & \lvert t \rvert > 0.5 \end{cases}$ | -| Convolution | $g(t)*h(t)=(g*h)(t)=\int_\infty^\infty g(\tau)h(t-\tau)d\tau$ | +```math +\begin{align*} + u(t) &= \begin{cases} 1, & t > 0 \\ \frac{1}{2}, & t = 0 \\ 0, & t < 0 \end{cases}&\text{Unit Step Function}\\ + \text{sgn}(t) &= \begin{cases} +1, & t > 0 \\ 0, & t = 0 \\ -1, & t < 0 \end{cases}&\text{Signum Function}\\ + \text{sinc}(2Wt) &= \frac{\sin(2\pi W t)}{2\pi W t}&\text{sinc Function}\\ + \text{rect}(t) = \Pi(t) &= \begin{cases} 1, & -0.5 < t < 0.5 \\ 0, & \lvert t \rvert > 0.5 \end{cases}&\text{Rectangular/Gate Function}\\ + g(t)*h(t)=(g*h)(t)&=\int_\infty^\infty g(\tau)h(t-\tau)d\tau&\text{Convolution}\\ +\end{align*} +``` ### Fourier transform of continuous time periodic signal @@ -73,21 +75,21 @@ Required for some questions on **sampling**: Transform a continuous time-periodic signal $x_p(t)=\sum_{n=-\infty}^\infty x(t-nT_s)$ with period $T_s$: -$$ +```math X_p(f)=\sum_{n=-\infty}^\infty C_n\delta(f-nf_s)\quad f_s=\frac{1}{T_s} -$$ +``` Calculate $C_n$ coefficient as follows from $x_p(t)$: -$$ +```math \begin{align*} % C_n&=X_p(nf_s)\\ C_n&=\frac{1}{T_s} \int_{T_s} x_p(t)\exp(-j2\pi f_s t)dt\\ &=\frac{1}{T_s} X(nf_s)\quad\color{red}\text{(TODO: Check)}\quad\color{white}\text{$x(t-nT_s)$ is contained in the interval $T_s$} \end{align*} -$$ +``` ### $\text{rect}$ function @@ -95,16 +97,16 @@ $$ ### Bessel function -$$ +```math \begin{align*} - \sum_{n\in\Z}{J_n}^2(\beta)&=1\\ + \sum_{n\in\mathbb{Z}}{J_n}^2(\beta)&=1\\ J_n(\beta)&=(-1)^nJ_{-n}(\beta) \end{align*} -$$ +``` ### White noise -$$ +```math \begin{align*} R_W(\tau)&=\frac{N_0}{2}\delta(\tau)=\frac{kT}{2}\delta(\tau)=\sigma^2\delta(\tau)\\ G_w(f)&=\frac{N_0}{2}\\ @@ -112,27 +114,27 @@ N_0&=kT\\ G_y(f)&=|H(f)|^2G_w(f)\\ G_y(f)&=G(f)G_w(f)\\ \end{align*} -$$ +``` ### WSS -$$ +```math \begin{align*} \mu_X(t) &= \mu_X\text{ Constant}\\ R_{XX}(t_1,t_2)&=R_X(t_1-t_2)=R_X(\tau)\\ E[X(t_1)X(t_2)]&=E[X(t)X(t+\tau)] \end{align*} -$$ +``` ### Ergodicity -$$ +```math \begin{align*} \braket{X(t)}_T&=\frac{1}{2T}\int_{-T}^{T}x(t)dt\\ \braket{X(t+\tau)X(t)}_T&=\frac{1}{2T}\int_{-T}^{T}x(t+\tau)x(t)dt\\ E[\braket{X(t)}_T]&=\frac{1}{2T}\int_{-T}^{T}x(t)dt=\frac{1}{2T}\int_{-T}^{T}m_Xdt=m_X\\ \end{align*} -$$ +``` | Type | Normal | Mean square sense | | ----------------------------------- | ------------------------------------------------------- | ----------------------------------------------------------- | @@ -143,17 +145,17 @@ $$ ### Other identities -$$ +```math \begin{align*} f*(g*h) &=(f*g)*h\quad\text{Convolution associative}\\ a(f*g) &= (af)*g \quad\text{Convolution associative}\\ \sum_{x=-\infty}^\infty(f(x a)\delta(\omega-x b))&=f\left(\frac{\omega a}{b}\right) \end{align*} -$$ +``` ### Other trig -$$ +```math \begin{align*} \cos2\theta=2 \cos^2 \theta-1&\Leftrightarrow\frac{\cos2\theta+1}{2}=\cos^2\theta\\ e^{-j\alpha}-e^{j\alpha}&=-2j \sin(\alpha)\\ @@ -164,11 +166,11 @@ $$ \sin(A-\pi/2)&=-\cos(A)\\ \cos(A-\pi/2)&=\sin(A)\\ \cos(A+\pi/2)&=-\sin(A)\\ - \int_{x\in\R}\text{sinc}(A x) &= \frac{1}{|A|}\\ + \int_{x\in\mathbb{R}}\text{sinc}(A x) &= \frac{1}{|A|}\\ \end{align*} -$$ +``` -$$ +```math \begin{align*} \cos(A+B) &= \cos (A) \cos (B)-\sin (A) \sin (B) \\ \sin(A+B) &= \sin (A) \cos (B)+\cos (A) \sin (B) \\ @@ -176,9 +178,9 @@ $$ \cos(A)\sin(B) &= \frac{1}{2} (\sin (A+B)-\sin (A-B)) \\ \sin(A)\sin(B) &= \frac{1}{2} (\cos (A-B)-\cos (A+B)) \\ \end{align*} -$$ +``` -$$ +```math \begin{align*} \cos(A)+\cos(B) &= 2 \cos \left(\frac{A}{2}-\frac{B}{2}\right) \cos \left(\frac{A}{2}+\frac{B}{2}\right) \\ \cos(A)-\cos(B) &= -2 \sin \left(\frac{A}{2}-\frac{B}{2}\right) \sin \left(\frac{A}{2}+\frac{B}{2}\right) \\ @@ -187,7 +189,7 @@ $$ \cos(A)+\sin(B)&= -2 \sin \left(\frac{A}{2}-\frac{B}{2}-\frac{\pi }{4}\right) \sin \left(\frac{A}{2}+\frac{B}{2}+\frac{\pi }{4}\right) \\ \cos(A)-\sin(B)&= -2 \sin \left(\frac{A}{2}+\frac{B}{2}-\frac{\pi }{4}\right) \sin \left(\frac{A}{2}-\frac{B}{2}+\frac{\pi }{4}\right) \\ \end{align*} -$$ +``` ## IQ/Complex envelope @@ -195,7 +197,7 @@ Def. $\tilde{g}(t)=g_I(t)+jg_Q(t)$ as the complex envelope. Best to convert to $ ### Convert complex envelope representation to time-domain representation of signal -$$ +```math \begin{align*} g(t)&=g_I(t)\cos(2\pi f_c t)-g_Q(t)\sin(2\pi f_c t)\\ &=\text{Re}[\tilde{g}(t)\exp{(j2\pi f_c t)}]\\ @@ -205,23 +207,23 @@ A(t)&=|g(t)|=\sqrt{g_I^2(t)+g_Q^2(t)}\quad\text{Amplitude}\\ g_I(t)&=A(t)\cos(\phi(t))\quad\text{In-phase component}\\ g_Q(t)&=A(t)\sin(\phi(t))\quad\text{Quadrature-phase component}\\ \end{align*} -$$ +``` ### For transfer function -$$ +```math \begin{align*} h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\ &=2\text{Re}[\tilde{h}(t)\exp{(j2\pi f_c t)}]\\ \Rightarrow\tilde{h}(t)&=h_I(t)/2+jh_Q(t)/2=A(t)/2\exp{(j\phi(t))} \end{align*} -$$ +``` ## AM ### CAM -$$ +```math \begin{align*} m_a &= \frac{\min_t|k_a m(t)|}{A_c} \quad\text{$k_a$ is the amplitude sensitivity ($\text{volt}^{-1}$), $m_a$ is the modulation index.}\\ m_a &= \frac{A_\text{max}-A_\text{min}}{A_\text{max}+A_\text{min}}\quad\text{ (Symmetrical $m(t)$)}\\ @@ -233,7 +235,7 @@ $$ \eta&=\frac{\text{Signal Power}}{\text{Total Power}}=\frac{P_x}{P_x+P_c}\\ B_T&=2f_m=2B \end{align*} -$$ +``` $B_T$: Signal bandwidth $B$: Bandwidth of modulating wave @@ -242,16 +244,16 @@ Overmodulation (resulting in phase reversals at crossing points): $m_a>1$ ### DSB-SC -$$ +```math \begin{align*} x_\text{DSB}(t) &= A_c \cos{(2\pi f_c t)} m(t)\\ B_T&=2f_m=2B \end{align*} -$$ +``` ## FM/PM -$$ +```math \begin{align*} s(t) &= A_c\cos\left[2\pi f_c t + k_p m(t)\right]\quad\text{Phase modulated (PM)}\\ s(t) &= A_c\cos\left[2\pi f_c t + 2 \pi k_f \int_0^t m(\tau) d\tau\right]\quad\text{Frequency modulated (FM)}\\ @@ -260,34 +262,30 @@ $$ \Delta f&=\beta f_m=k_f A_m f_m = \max_t(k_f m(t))- \min_t(k_f m(t))\quad\text{Maximum frequency deviation}\\ D&=\frac{\Delta f}{W_m}\quad\text{Deviation ratio, where $W_m$ is bandwidth of $m(t)$ (Use FT)} \end{align*} -$$ +``` ### Bessel form and magnitude spectrum (single tone) -$$ +```math \begin{align*} s(t) &= A_c\cos\left[2\pi f_c t + \beta \sin(2\pi f_m t)\right] \Leftrightarrow s(t)= A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[2\pi(f_c+nf_m)t] \end{align*} -$$ - - +``` ### FM signal power -$$ +```math \begin{align*} P_\text{av}&=\frac{{A_c}^2}{2}\\ P_\text{band\_index}&=\frac{{A_c}^2{J_\text{band\_index}}^2(\beta)}{2}\\ \text{band\_index}&=0\implies f_c+0f_m\\ \text{band\_index}&=1\implies f_c+1f_m,\dots\\ \end{align*} -$$ +``` ### Carson's rule to find $B$ (98% power bandwidth rule) -$$ +```math \begin{align*} B &= 2Mf_m = 2(\beta + 1)f_m\\ &= 2(\Delta f+f_m)\\ @@ -298,7 +296,7 @@ B &= \begin{cases} 2(\Delta\phi + 1)f_m & \text{PM, sinusoidal message} \end{cases}\\ \end{align*} -$$ +``` #### $\Delta f$ of arbitrary modulating signal @@ -306,7 +304,7 @@ Find instantaneous frequency $f_\text{FM}$. $M$: Number of **pairs** of significant sidebands -$$ +```math \begin{align*} s(t)&=A_c\cos(\theta_\text{FM}(t))\\ f_\text{FM}(t) &= \frac{1}{2\pi}\frac{d\theta_\text{FM}(t)}{dt}\\ @@ -317,17 +315,17 @@ W_m &= \text{max}(\text{frequencies in $\theta_\text{FM}(t)$...}) \\ D &= \frac{\Delta f}{W_m}\\ B_T &= 2(D+1)W_m \end{align*} -$$ +``` ### Complex envelope -$$ +```math \begin{align*} s(t)&=A_c\cos(2\pi f_c t+\beta\sin(2\pi f_m t)) \Leftrightarrow \tilde{s}(t) = A_c\exp(j\beta\sin(2\pi f_m t))\\ s(t)&=\text{Re}[\tilde{s}(t)\exp{(j2\pi f_c t)}]\\ \tilde{s}(t) &= A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\exp(j2\pi f_m t) \end{align*} -$$ +``` ### Band @@ -337,7 +335,7 @@ $$ ## Power, energy and autocorrelation -$$ +```math \begin{align*} G_\text{WGN}(f)&=\frac{N_0}{2}\\ G_x(f)&=|H(f)|^2G_w(f)\text{ (PSD)}\\ @@ -350,24 +348,24 @@ $$ E&=\int_{-\infty}^{\infty}|x(t)|^2dt=|X(f)|^2\\ R_x(\tau) &= \mathfrak{F}(G_x(f))\quad\text{PSD to Autocorrelation} \end{align*} -$$ +``` ## ## Noise performance -$$ +```math \begin{align*} \text{CNR}_\text{in} &= \frac{P_\text{in}}{P_\text{noise}}\\ \text{CNR}_\text{in,FM} &= \frac{A^2}{2WN_0}\\ \text{SNR}_\text{FM} &= \frac{3A^2k_f^2P}{2N_0W^3}\\ \text{SNR(dB)} &= 10\log_{10}(\text{SNR}) \quad\text{Decibels from ratio} \end{align*} -$$ +``` ## Sampling -$$ +```math \begin{align*} t&=nT_s\\ T_s&=\frac{1}{f_s}\\ @@ -375,7 +373,7 @@ $$ X_s(f)&=X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-\frac{n}{T_s}\right)=X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-n f_s\right)\\ B&>\frac{1}{2}f_s, 2B>f_s\rightarrow\text{Aliasing}\\ \end{align*} -$$ +``` ### Procedure to reconstruct sampled signal @@ -397,21 +395,21 @@ Required for some questions on **sampling**: Transform a continuous time-periodic signal $x_p(t)=\sum_{n=-\infty}^\infty x(t-nT_s)$ with period $T_s$: -$$ +```math X_p(f)=\sum_{n=-\infty}^\infty C_n\delta(f-nf_s)\quad f_s=\frac{1}{T_s} -$$ +``` Calculate $C_n$ coefficient as follows from $x_p(t)$: -$$ +```math \begin{align*} % C_n&=X_p(nf_s)\\ C_n&=\frac{1}{T_s} \int_{T_s} x_p(t)\exp(-j2\pi f_s t)dt\\ &=\frac{1}{T_s} X(nf_s)\quad\color{red}\text{(TODO: Check)}\quad\color{white}\text{$x(t-nT_s)$ is contained in the interval $T_s$} \end{align*} -$$ +``` @@ -428,15 +426,15 @@ Do not transmit more than $2B$ samples per second over a channel of $B$ bandwidt ## Quantizer -$$ +```math \begin{align*} \Delta &= \frac{x_\text{Max}-x_\text{Min}}{2^k} \quad\text{for $k$-bit quantizer (V/lsb)}\\ \end{align*} -$$ +``` ### Quantization noise -$$ +```math \begin{align*} e &:= y-x\quad\text{Quantization error}\\ \mu_E &= E[E] = 0\quad\text{Zero mean}\\ @@ -444,7 +442,7 @@ $$ \text{SQNR}&=\frac{\text{Signal power}}{\text{Quantization noise}}\\ \text{SQNR(dB)}&=10\log_{10}(\text{SQNR}) \end{align*} -$$ +``` ### Insert here figure 8.17 from M F Mesiya - Contemporary Communication Systems (Add image to `images/quantizer.png`) @@ -455,7 +453,7 @@ $$ ![binary_codes](images/Line_Codes.drawio.svg) -$$ +```math \begin{align*} R_b&\rightarrow\text{Bit rate}\\ D&\rightarrow\text{Symbol rate | }R_d\text{ | }1/T_b\\ @@ -469,7 +467,7 @@ $$ G_\text{unipolarNRZ}(f)&=\frac{A^2}{4R_b}\left(\text{sinc}^2\left(\frac{f}{R_b}\right)+R_b\delta(f)\right)\\ G_\text{unipolarRZ}(f)&=\frac{A^2}{16} \left(\sum _{l=-\infty }^{\infty } \delta \left(f-\frac{l}{T_b}\right) \left| \text{sinc}(\text{duty} \times l) \right| {}^2+T_b \left| \text{sinc}\left(\text{duty} \times f T_b\right) \right| {}^2\right), \text{NB}_0=2R_b \end{align*} -$$ +``` ## Modulation and basis functions @@ -479,25 +477,27 @@ $$ #### Basis functions -$$ +```math \begin{align*} \varphi_1(t) &= \sqrt{\frac{2}{T_b}}\cos(2\pi f_c t)\quad0\leq t\leq T_b\\ \end{align*} -$$ +``` #### Symbol mapping -$$b_n:\{1,0\}\to a_n:\{1,0\}$$ +```math +b_n:\{1,0\}\to a_n:\{1,0\} +``` #### 2 possible waveforms -$$ +```math \begin{align*} s_1(t)&=A_c\sqrt{\frac{T_b}{2}}\varphi_1(t)=\sqrt{2E_b}\varphi_1(t)\\ s_1(t)&=0\\ &\text{Since $E_b=E_\text{average}=\frac{1}{2}(\frac{{A_c}^2}{2}\times T_b + 0)=\frac{{A_c}^2}{4}T_b$} \end{align*} -$$ +``` Distance is $d=\sqrt{2E_b}$ @@ -505,25 +505,27 @@ Distance is $d=\sqrt{2E_b}$ #### Basis functions -$$ +```math \begin{align*} \varphi_1(t) &= \sqrt{\frac{2}{T_b}}\cos(2\pi f_c t)\quad0\leq t\leq T_b\\ \end{align*} -$$ +``` #### Symbol mapping -$$b_n:\{1,0\}\to a_n:\{1,\color{lime}-1\color{white}\}$$ +```math +b_n:\{1,0\}\to a_n:\{1,\color{lime}-1\color{white}\} +``` #### 2 possible waveforms -$$ +```math \begin{align*} s_1(t)&=A_c\sqrt{\frac{T_b}{2}}\varphi_1(t)=\sqrt{E_b}\varphi_1(t)\\ s_1(t)&=-A_c\sqrt{\frac{T_b}{2}}\varphi_1(t)=-\sqrt{E_b}\varphi_2(t)\\ &\text{Since $E_b=E_\text{average}=\frac{1}{2}(\frac{{A_c}^2}{2}\times T_b + \frac{{A_c}^2}{2}\times T_b)=\frac{{A_c}^2}{2}T_b$} \end{align*} -$$ +``` Distance is $d=2\sqrt{E_b}$ @@ -531,45 +533,45 @@ Distance is $d=2\sqrt{E_b}$ #### Basis functions -$$ +```math \begin{align*} T &= 2 T_b\quad\text{Time per symbol for two bits $T_b$}\\ \varphi_1(t) &= \sqrt{\frac{2}{T}}\cos(2\pi f_c t)\quad0\leq t\leq T\\ \varphi_2(t) &= \sqrt{\frac{2}{T}}\sin(2\pi f_c t)\quad0\leq t\leq T\\ \end{align*} -$$ +``` ### 4 possible waveforms -$$ +```math \begin{align*} s_1(t)&=\sqrt{E_s/2}\left[\varphi_1(t)+\varphi_2(t)\right]\\ s_2(t)&=\sqrt{E_s/2}\left[\varphi_1(t)-\varphi_2(t)\right]\\ s_3(t)&=\sqrt{E_s/2}\left[-\varphi_1(t)+\varphi_2(t)\right]\\ s_4(t)&=\sqrt{E_s/2}\left[-\varphi_1(t)-\varphi_2(t)\right]\\ \end{align*} -$$ +``` Note on energy per symbol: Since $|s_i(t)|=A_c$, have to normalize distance as follows: -$$ +```math \begin{align*} s_i(t)&=A_c\sqrt{T/2}/\sqrt{2}\times\left[\alpha_{1i}\varphi_1(t)+\alpha_{2i}\varphi_2(t)\right]\\ &=\sqrt{T{A_c}^2/4}\left[\alpha_{1i}\varphi_1(t)+\alpha_{2i}\varphi_2(t)\right]\\ &=\sqrt{E_s/2}\left[\alpha_{1i}\varphi_1(t)+\alpha_{2i}\varphi_2(t)\right]\\ \end{align*} -$$ +``` #### Signal -$$ +```math \begin{align*} \text{Symbol mapping: }& \left\{1,0\right\}\to\left\{1,-1\right\}\\ I(t) &= b_{2n}\varphi_1(t)\quad\text{Even bits}\\ Q(t) &= b_{2n+1}\varphi_2(t)\quad\text{Odd bits}\\ x(t) &= A_c[I(t)\cos(2\pi f_c t)-Q(t)\sin(2\pi f_c t)] \end{align*} -$$ +``` ### Example of waveform @@ -615,20 +617,20 @@ Remember that $T=2T_b$ Find transfer function $h(t)$ of matched filter and apply to an input: -$$ +```math \begin{align*} h(t)&=s_1(T-t)-s_2(T-t)\\ h(t)&=s^*(T-t) \qquad\text{((.)* is the conjugate)}\\ s_{on}(t)&=h(t)*s_n(t)=\int_\infty^\infty h(\tau)s_n(t-\tau)d\tau\quad\text{Filter output}\\ n_o(t)&=h(t)*n(t)\quad\text{Noise at filter output} \end{align*} -$$ +``` ### 2. Bit error rate Bit error rate (BER) from matched filter outputs and filter output noise -$$ +```math \begin{align*} % H_\text{opt}(f)&=\max_{H(f)}\left(\frac{s_{o1}-s_{o2}}{2\sigma_o}\right) @@ -643,7 +645,7 @@ $$ \text{BER}_\text{unipolarNRZ|BASK}&=Q\left(\sqrt{\frac{d^2}{N_0}}\right)=Q\left(\sqrt{\frac{E_b}{N_0}}\right)\\ \text{BER}_\text{polarNRZ|BPSK}&=Q\left(\sqrt{\frac{2d^2}{N_0}}\right)=Q\left(\sqrt{\frac{2E_b}{N_0}}\right)\\ \end{align*} -$$ +```
    @@ -673,7 +675,7 @@ $$ | $x$ | $Q(x)$ | $x$ | $Q(x)$ | $x$ | $Q(x)$ | $x$ | $Q(x)$ | | ------ | ---------- | ------ | ----------------------- | ------ | ------------------------ | ------ | ------------------------ | -| $0.00$ | $0.5 | $2.30$ | $0.010724$ | $4.55$ | $2.6823 \times 10^{-6}$ | $6.80$ | $5.231 \times 10^{-12}$ | +| $0.00$ | $0.5$ | $2.30$ | $0.010724$ | $4.55$ | $2.6823 \times 10^{-6}$ | $6.80$ | $5.231 \times 10^{-12}$ | | $0.05$ | $0.48006$ | $2.35$ | $0.0093867$ | $4.60$ | $2.1125 \times 10^{-6}$ | $6.85$ | $3.6925 \times 10^{-12}$ | | $0.10$ | $0.46017$ | $2.40$ | $0.0081975$ | $4.65$ | $1.6597 \times 10^{-6}$ | $6.90$ | $2.6001 \times 10^{-12}$ | | $0.15$ | $0.44038$ | $2.45$ | $0.0071428$ | $4.70$ | $1.3008 \times 10^{-6}$ | $6.95$ | $1.8264 \times 10^{-12}$ | @@ -726,7 +728,7 @@ Adapted from table 6.1 M F Mesiya - Contemporary Communication Systems ### Receiver output shit -$$ +```math \begin{align*} r_o(t)&=\begin{cases} s_{o1}(t)+n_o(t) & \text{code 1}\\ @@ -734,14 +736,14 @@ $$ \end{cases}\\ n&: \text{AWGN with }\sigma_o^2\\ \end{align*} -$$ +``` +``` --> ## ISI, channel model @@ -751,20 +753,22 @@ TODO: ### Nomenclature -$$ +```math \begin{align*} D&\rightarrow\text{Symbol Rate, Max. Signalling Rate}\\ T&\rightarrow\text{Symbol Duration}\\ M&\rightarrow\text{Symbol set size}\\ W&\rightarrow\text{Bandwidth}\\ \end{align*} -$$ +``` ### Raised cosine (RC) pulse ![Raised cosine pulse](images/RC.drawio.svg) -$$0\leq\alpha\leq1$$ +```math +0\leq\alpha\leq1 +``` ⚠ NOTE might not be safe to assume $T'=T$, if you can solve the question without $T$ then use that method. @@ -773,13 +777,6 @@ To solve this type of question: 1. Use the formula for $D$ below 2. Consult the BER table below to get the BER which relates the noise of the channel $N_0$ to $E_b$ and to $R_b$. - - | Linear modulation ($M$-PSK, $M$-QAM) | NRZ unipolar encoding | | --------------------------------------------------- | -------------------------------------------------- | | $W=B_\text{\color{lime}abs-abs}$ | $W=B_\text{\color{lime}abs}$ | @@ -788,35 +785,35 @@ $$ --> #### Symbol set size $M$ -$$ +```math \begin{align*} D\text{ symbol/s}&=\frac{2W\text{ Hz}}{1+\alpha}\\ R_b\text{ bit/s}&=(D\text{ symbol/s})\times(k\text{ bit/symbol})\\ M\text{ symbol/set}&=2^k\\ E_b&=PT=P_\text{av}/R_b\quad\text{Energy per bit}\\ \end{align*} -$$ +``` ### Nyquist stuff -#### Condition for 0 ISI +#### Condition for 0 ISI TODO: -$$ +```math P_r(kT)=\begin{cases} 1 & k=0\\ 0 & k\neq0 \end{cases} -$$ +``` #### Other -$$ +```math \begin{align*} \text{Excess BW}&=B_\text{abs}-B_\text{Nyquist}=\frac{1+\alpha}{2T}-\frac{1}{2T}=\frac{\alpha}{2T}\quad\text{FOR NRZ (Use correct $B_\text{abs}$)}\\ \alpha&=\frac{\text{Excess BW}}{B_\text{Nyquist}}=\frac{B_\text{abs}-B_\text{Nyquist}}{B_\text{Nyquist}}\\ T&=1/D \end{align*} -$$ +```
    @@ -854,7 +851,7 @@ Adapted from table 11.4 M F Mesiya - Contemporary Communication Systems ### Entropy for discrete random variables -$$ +```math \begin{align*} H(x) &\geq 0\\ H(x) &= -\sum_{x_i\in A_x} p_X(x_i) \log_2(p_X(x_i))\\ @@ -866,7 +863,7 @@ $$ H(x|y) &= H(x,y)-H(y)\\ H(x,y) &= H(x) + H(y|x) = H(y) + H(x|y)\\ \end{align*} -$$ +``` Entropy is **maximized** when all have an equal probability. @@ -874,64 +871,63 @@ Entropy is **maximized** when all have an equal probability. TODO: Cut out if not required -$$ +```math \begin{align*} h(x) &= -\int_\mathbb{R}f_X(x)\log_2(f_X(x))dx \end{align*} -$$ +``` ### Mutual information Amount of entropy decrease of $x$ after observation by $y$. -$$ +```math \begin{align*} I(x;y) &= H(x)-H(x|y)=H(y)-H(y|x)\\ \end{align*} -$$ +``` ### Channel model Vertical, $x$: input\ Horizontal, $y$: output -$$ +```math \mathbf{P}=\left[\begin{matrix} p_{11} & p_{12} &\dots & p_{1N}\\ p_{21} & p_{22} &\dots & p_{2N}\\ \vdots & \vdots &\ddots & \vdots\\ p_{M1} & p_{M2} &\dots & p_{MN}\\ \end{matrix}\right] -$$ +``` -$$ +```math \begin{array}{c|cccc} - P(y_j|x_i)& y_1 & y_2 & \dots & y_N \\ - \hline + P(y_j|x_i)& y_1 & y_2 & \dots & y_N \\\hline x_1 & p_{11} & p_{12} & \dots & p_{1N} \\ x_2 & p_{21} & p_{22} & \dots & p_{2N} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_M & p_{M1} & p_{M2} & \dots & p_{MN} \\ \end{array} -$$ +``` Input has probability distribution $p_X(a_i)=P(X=a_i)$ -Channel maps alphabet $\{a_1,\dots,a_M\} \to \{b_1,\dots,b_N\}$ +Channel maps alphabet $`\{a_1,\dots,a_M\} \to \{b_1,\dots,b_N\}`$ Output has probabiltiy distribution $p_Y(b_j)=P(y=b_j)$ -$$ +```math \begin{align*} p_Y(b_j) &= \sum_{i=1}^{M}P[x=a_i,y=b_j]\quad 1\leq j\leq N \\ &= \sum_{i=1}^{M}P[X=a_i]P[Y=b_j|X=a_i]\\ [\begin{matrix}p_Y(b_0)&p_Y(b_1)&\dots&p_Y(b_j)\end{matrix}] &= [\begin{matrix}p_X(a_0)&p_X(a_1)&\dots&p_X(a_i)\end{matrix}]\times\mathbf{P} \end{align*} -$$ +``` #### Fast procedure to calculate $I(y;x)$ -$$ +```math \begin{align*} &\text{1. Find }H(x)\\ &\text{2. Find }[\begin{matrix}p_Y(b_0)&p_Y(b_1)&\dots&p_Y(b_j)\end{matrix}] = [\begin{matrix}p_X(a_0)&p_X(a_1)&\dots&p_X(a_i)\end{matrix}]\times\mathbf{P}\\ @@ -940,7 +936,7 @@ $$ &\text{5. Find }H(x|y)=H(x,y)-H(y)\\ &\text{6. Find }I(y;x)=H(x)-H(x|y)\\ \end{align*} -$$ +``` ### Channel types @@ -951,7 +947,7 @@ $$ #### Channel capacity of weakly symmetric channel -$$ +```math \begin{align*} C &\to\text{Channel capacity (bits/channels used)}\\ N &\to\text{Output alphabet size}\\ @@ -959,38 +955,38 @@ $$ C &= \log_2(N)-H(\mathbf{p})\quad\text{Capacity for weakly symmetric and symmetric channels}\\ R &< C \text{ for error-free transmission} \end{align*} -$$ +``` #### Channel capacity of an AWGN channel -$$ +```math y_i=x_i+n_i\quad n_i\thicksim N(0,N_0/2) -$$ +``` -$$ +```math C=\frac{1}{2}\log_2\left(1+\frac{P_\text{av}}{N_0/2}\right) -$$ +``` #### Channel capacity of a bandwidth AWGN channel Note: Define XOR ($\oplus$) as exclusive OR, or modulo-2 addition. -$$ +```math \begin{align*} P_s&\to\text{Bandwidth limited average power}\\ y_i&=\text{bandpass}_W(x_i)+n_i\quad n_i\thicksim N(0,N_0/2)\\ C&=W\log_2\left(1+\frac{P_s}{N_0 W}\right)\\ C&=W\log_2(1+\text{SNR})\quad\text{SNR}=P_s/(N_0 W) \end{align*} -$$ +``` ## Channel code -| | | | -| ---------------- | --------------------------------- | -------------------------------------------------------------------------------------------------------------------------- | -| Hamming weight | $w_H(x)$ | Number of `'1'` in codeword $x$ | -| Hamming distance | $d_H(x_1,x_2)=w_H(x_1\oplus x_2)$ | Number of different bits between codewords $x_1$ and $x_2$ which is the hamming weight of the XOR of the two codes. | -| Minimum distance | $d_\text{min}$ | **IMPORTANT**: $x\neq\bold{0}$, excludes weight of all-zero codeword. For a linear block code, $d_\text{min}=w_\text{min}$ | +| | | | +| ---------------- | --------------------------------- | ---------------------------------------------------------------------------------------------------------------------------- | +| Hamming weight | $w_H(x)$ | Number of `'1'` in codeword $x$ | +| Hamming distance | $d_H(x_1,x_2)=w_H(x_1\oplus x_2)$ | Number of different bits between codewords $x_1$ and $x_2$ which is the hamming weight of the XOR of the two codes. | +| Minimum distance | $d_\text{min}$ | **IMPORTANT**: $x\neq\textbf{0}$, excludes weight of all-zero codeword. For a linear block code, $d_\text{min}=w_\text{min}$ | ### Linear block code @@ -1011,7 +1007,7 @@ For a linear block code, $d_\text{min}=w_\text{min}$ Each generator vector is a binary string of size $n$. There are $k$ generator vectors in $\mathbf{G}$. -$$ +```math \begin{align*} \mathbf{g}_i&=[\begin{matrix} g_{i,0}& \dots & g_{i,n-2} & g_{i,n-1} @@ -1029,11 +1025,11 @@ $$ g_{k-1,0}& \dots & g_{k-1,n-2} & g_{k-1,n-1}\\ \end{matrix}\right] \end{align*} -$$ +``` A message block $\mathbf{m}$ is coded as $\mathbf{x}$ using the generation codewords in $\mathbf{G}$: -$$ +```math \begin{align*} \mathbf{m}&=[\begin{matrix} m_{0}& \dots & m_{n-2} & m_{k-1} @@ -1041,13 +1037,13 @@ $$ \color{darkgray}\mathbf{m}&\color{darkgray}=[101001]\quad\text{Example for $k=6$}\\ \mathbf{x} &= \mathbf{m}\mathbf{G}=m_0\mathbf{g}_0+m_1\mathbf{g}_1+\dots+m_{k-1}\mathbf{g}_{k-1} \end{align*} -$$ +``` ### Systemic linear block code Contains $k$ message bits (Copy $\mathbf{m}$ as-is) and $(n-k)$ parity bits after the message bits. -$$ +```math \begin{align*} \mathbf{G}&=\begin{array}{c|c}[\mathbf{I}_k & \mathbf{P}]\end{array}=\left[ \begin{array}{c|c} @@ -1070,13 +1066,13 @@ $$ \mathbf{x} &= \mathbf{m}\mathbf{G}= \mathbf{m} \begin{array}{c|c}[\mathbf{I}_k & \mathbf{P}]\end{array}=\begin{array}{c|c}[\mathbf{mI}_k & \mathbf{mP}]\end{array}=\begin{array}{c|c}[\mathbf{m} & \mathbf{b}]\end{array}\\ \mathbf{b} &= \mathbf{m}\mathbf{P}\quad\text{Parity bits of $\mathbf{x}$} \end{align*} -$$ +``` #### Parity check matrix $\mathbf{H}$ Transpose $\mathbf{P}$ for the parity check matrix -$$ +```math \begin{align*} \mathbf{H}&=\begin{array}{c|c}[\mathbf{P}^\text{T} & \mathbf{I}_{n-k}]\end{array}\\ &=\left[ @@ -1103,7 +1099,7 @@ $$ \end{matrix}\end{array}\right]\\ \mathbf{xH}^\text{T}&=\mathbf{0}\implies\text{Codeword is valid} \end{align*} -$$ +``` #### Procedure to find parity check matrix from list of codewords @@ -1114,30 +1110,28 @@ $$ Example: -$$ +```math \begin{array}{cccc} - x_1 & x_2 & x_3 & x_4 & x_5 \\ - \hline + x_1 & x_2 & x_3 & x_4 & x_5 \\\hline \color{magenta}1&\color{magenta}0&1&1&0\\ \color{magenta}0&\color{magenta}1&1&1&1\\ \color{magenta}0&\color{magenta}0&0&0&0\\ \color{magenta}1&\color{magenta}1&0&0&1\\ \end{array} -$$ +``` Set $x_1,x_2$ as information bits. Express $x_3,x_4,x_5$ in terms of $x_1,x_2$. -$$ -\begin{align*} +```math \begin{align*} +\begin{aligned} x_3 &= x_1\oplus x_2\\ x_4 &= x_1\oplus x_2\\ x_5 &= x_2\\ -\end{align*} +\end{aligned} \implies\textbf{P}&= -\begin{array}{c|ccc} - & x_1 & x_2 \\ - \hline +\begin{array}{c|cc} + & x_1 & x_2 \\\hline x_3&1&1&\\ x_4&1&1&\\ x_5&0&1&\\ @@ -1156,7 +1150,7 @@ $$ 0 & 0 & 1\\ \end{matrix}\end{array}\right] \end{align*} -$$ +``` #### Error detection and correction @@ -1166,6 +1160,6 @@ $$ ## CHECKLIST -- Transfer function in complex envelope form ($\tilde{h}(t)$) should be divided by two. +- Transfer function in complex envelope form $\tilde{h}(t)$ should be divided by two. - Convolutions: do not forget width when using graphical method - todo: add more items to check diff --git a/README.pdf b/README.pdf new file mode 100644 index 0000000..03b4980 Binary files /dev/null and b/README.pdf differ